Further Pure Math (Trigonometry 1)

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1. (2013/jan/paper02/q2)

Using the identities $\quad \sin (A+B)=\sin A \cos B+\cos A \sin B$

$$\begin{gathered}\cos (A+B)=\cos A \cos B-\sin A \sin B \\\tan A=\frac{\sin A}{\cos A}\end{gathered}$$

(a) show that $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$

(b) Hence show that

(i) $\tan 105^{\circ}=\frac{1+\sqrt{3}}{1-\sqrt{3}}$

(ii) $\tan 15^{\circ}=\frac{\sqrt{3}-1}{1+\sqrt{3}}$ (4)


2. $(2017 /$ june $/$ paper01 $/ \mathrm{q} 4)$

Solve, for $0 \leqslant \theta<\pi$, to 4 significant figures,

(a) $(\tan \theta-3)(\tan \theta+2)=0$ (3)

(b) $6 \cos ^{2} \theta-\sin \theta=5$


3. (2012/jan/paper02/q8)

$$\begin{gathered}\sin (A+B)=\sin A \cos B+\cos A \sin B \\\cos (A+B)=\cos A \cos B-\sin A \sin B \\\tan A=\frac{\sin A}{\cos A}\end{gathered}$$

(a) Show that $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$ (3)

(b) Hence write down an expression for $\tan 2 \theta$ in terms of $\tan \theta$

(c) Show that $\tan 3 \theta=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta}$ (4)

Given that $\tan 3 \theta=-1$ and $\tan \theta \neq \pm \frac{\sqrt{3}}{3}$

(d) without finding the value of $\theta$, show that $\tan ^{3} \theta+3 \tan ^{2} \theta-3 \tan \theta-1=0$

Given also that $\tan \theta \neq 1$

(e) find the exact values of tan $\theta$, giving your answers in the form $a \pm \sqrt{b}$ where $a$ and $b$ are integers.


4. (2012/june/рарег01/q7)

$$\cos (A+B)=\cos A \cos B-\sin A \sin B$$

(a) Express $\cos \left(2 x+45^{\circ}\right)$ in the form $M \cos 2 x+N \sin 2 x$, where $M$ and $N$ are constants, giving the exact value of $M$ and the exact value of $N$

(b) Solve, for $0^{\circ} \leqslant x \leqslant 180^{\circ}$, the equation $\cos 2 x-\sin 2 x=1$

The maximum value of $\cos 2 x-\sin 2 x$ is $k$.

(c) Find the exact value of $k$

(d) Find the smallest positive value of $x$ for which a maximum occurs.


5. $\left(2013 / \mathrm{jan} / \mathrm{Paper01} / \mathrm{q8}\right)$

Solve, for $0 \leqslant \theta \leqslant \pi$, giving each solution to 3 significant figures,

(a) $5 \sin \theta-1=0$ (3)

(b) $\tan \left(2 \theta+\frac{\pi}{3}\right)=0.4$ (4)

(c) $4 \sin ^{2} \theta-7 \cos \theta=2$ (4)


6. (2013/june/paper01/q4)

Solve, for $-90<x \leqslant 90$, the equation

$$6 \sin ^{2} x^{\circ}-\cos x^{\circ}-4=0$$


7. (2014/jan/paper01/q8)

$$\begin{gathered}\sin (A+B)=\sin A \cos B+\cos A \sin B \\\tan A=\frac{\sin A}{\cos A}\end{gathered}$$

(a) Show that the equation

$$4 \sin (x+\alpha)=7 \sin (x-\alpha)$$

can be written in the form

$$3 \tan x=11 \tan \alpha$$

(b) Hence solve, to 1 decimal place,

$$4 \sin (3 y+45)^{\circ}=7 \sin (3 y-45)^{\circ} \text { for } 0 \leqslant y \leqslant 180$$


8. (2014/june/paper01/q2)

Solve, in degrees to 1 decimal place, for $0 \leqslant \theta<180^{\circ}$

(a) $\tan 2 \theta=1.5$

(b) $(3 \cos \theta+1)(2 \cos \theta+3)=-2$


9. (2014/june/paper02/q10)

Using the identities $\quad \cos (A+B)=\cos A \cos B-\sin A \sin B$

$$\sin (A+B)=\sin A \cos B+\cos A \sin B$$

(a) (i) show that $\cos 2 A=1-2 \sin ^{2} A$

(ii) write down an expression for $\sin 2 A$ in terms of $\sin A$ and $\cos A$

(b) Hence show that $\sin 3 A=3 \sin A-4 \sin ^{3} A$

(c) Solve, for $0 \leqslant x \leqslant \pi$, the equation $16 \sin ^{3} x-12 \sin x+1=0$

Give your answers correct to 3 significant figures.

(d) Find $\displaystyle\int\left(24 \sin ^{3} \theta+6 \cos \theta\right) \mathrm{d} \theta$ (2)

(e) Hence evaluate $\displaystyle\int_{0}^{\frac{\pi}{3}}\left(24 \sin ^{3} \theta+6 \cos \theta\right) \mathrm{d} \theta$, giving your answer in the form $a+b \sqrt{c}$, where $a, b$ and $c$ are integers.


10. ( $2015 / \mathrm{jan} / \mathrm{paper} 01 / \mathrm{q} 8)$

$$\tan \theta=\frac{\sin \theta}{\cos \theta}$$

(a) Using the above identity, show that $1+\tan ^{2} \theta=\frac{1}{\cos ^{2} \theta}$

(b) Show that $\frac{1+\sin \theta \cos \theta+\sin ^{2} \theta}{\cos ^{2} \theta}=1+\tan \theta+2 \tan ^{2} \theta$

(c) Solve the equation $1+\sin \theta \cos \theta+\sin ^{2} \theta=4 \cos ^{2} \theta$ for $0 \leqslant \theta \leqslant 180^{\circ}$. Give your answers to 1 decimal place, where appropriate.


11. ( $2015 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 4)$

$$\begin{aligned}&\sin (A+B)=\sin A \cos B+\cos A \sin B \\&\cos (A+B)=\cos A \cos B-\sin A \sin B\end{aligned}$$

(a) Write down the exact value of $\sin 45^{\circ}$

Given that $\sin \theta=\frac{\sqrt{5}}{2 \sqrt{2}}$ and $\cos \theta=\frac{\sqrt{3}}{2 \sqrt{2}}$

(b) show that $\sin \left(45^{\circ}+\theta\right)=\frac{\sqrt{3}+\sqrt{5}}{4}$

(c) Find the exact value of $\cos \left(45^{\circ}+\theta\right)$

(d) Show that $\sin \left(45^{\circ}+\theta\right) \times \cos \left(45^{\circ}+\theta\right)=-\frac{1}{8}$


12. (2015/june/paper01/q8)

Using the identities $\cos (A+B)=\cos A \cos B-\sin A \sin B$

$$\sin (A+B)=\sin A \cos B+\cos A \sin B$$

(a) (i) show that $\cos 2 A=1-2 \sin ^{2} A$

(ii) express $\sin 2 A$ in terms of $\sin A$ and $\cos A$, simplifying your answer.(1)

(b) Hence show that $\sin 3 A=3 \sin A-4 \sin ^{3} A$(4)

(c) Solve, for $-90^{\circ} \leqslant A \leqslant 90^{\circ}$, the equation

$$8 \sin ^{3} A-6 \sin A=1$$

(d) (i) Find $\displaystyle\int \sin ^{3} \theta \mathrm{d} \theta$

(ii) Evaluate $\displaystyle\int_{0}^{\frac{\pi}{4}} \sin ^{3} \theta \mathrm{d} \theta$, giving your answer in the form $\frac{a-b \sqrt{2}}{c}$, where $a, b$, and $c$ are integers. $(5)$


13. (2016/jan/paper01/q6)

Giving your solutions to 3 decimal places, solve the equation

(a) $\cos x=0.4$ $-\pi<x<\pi$

(b) $\tan \left(2 \theta+\frac{\pi}{4}\right)=1.5 \quad 0<\theta<\pi$ (4)


14. (2016/jan/paper02/q6)

$$\begin{aligned}\sin (A+B) &=\sin A \cos B+\cos A \sin B \\\cos (A+B) &=\cos A \cos B-\sin A \sin B \\\frac{\sin A}{\cos A} &=\tan A\end{aligned}$$

Using the above formulae, show that

(a) $\sin 2 x=2 \sin x \cos x$

(b) $\quad \cos 2 x=\cos ^{2} x-\sin ^{2} x$

(c) $\frac{\sin 2 x}{1+\cos 2 x}=\tan x$


15. (2016/june/paper01/q5)

Using the identities

$$\begin{gathered}\sin (A+B)=\sin A \cos B+\cos A \sin B \\\tan A=\frac{\sin A}{\cos A}\end{gathered}$$

(a) show that the equation

$$3 \sin (x+\alpha)=5 \sin (x-\alpha)$$

can be written in the form $\tan x=4 \tan \alpha$

(b) Hence solve, to the nearest integer, the equation

$$3 \sin (2 y+30)^{\circ}=5 \sin (2 y-30)^{\circ} \quad \text { for } 90 \leqslant y<180$$


16. (2016/june/paper02/q9)

$$\begin{aligned}&\sin (A+B)=\sin A \cos B+\cos A \sin B \\&\cos (A+B)=\cos A \cos B-\sin A \sin B\end{aligned}$$

Using the above identities

(a) show that $\cos 2 \theta=2 \cos ^{2} \theta-1$ (3)

(b) find a simplified expression for $\sin 2 \theta$ in terms of $\sin \theta$ and $\cos \theta$

(c) show that $\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta$

Hence, or otherwise,

(d) solve, for $0 \leqslant \theta<\pi$ giving your answers in terms of $\pi$, the equation

$$6 \cos \theta-8 \cos ^{3} \theta+1=0$$

(e) find

(i) $\displaystyle\int\left(8 \cos ^{3} \theta+4 \sin \theta\right) \mathrm{d} \theta$

(ii) the exact value of $\displaystyle\int_{0}^{\frac{x}{3}}\left(8 \cos ^{3} \theta+4 \sin \theta\right) \mathrm{d} \theta$


17. (2017/jan/рареr02/q2)

(a) Show that the equation $6 \cos ^{2} \alpha-\sin \alpha=5$ can be written as

$$6 \sin ^{2} \alpha+\sin \alpha-1=0$$

(b) Solve, to 1 decimal place where appropriate, for $0 \leqslant \theta \leqslant 90$

$$6 \cos ^{2}(2 \theta+40)^{\circ}-\sin (2 \theta+40)^{\circ}=5$$


18. (2017/jan/рареr02/q4)

$$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$$

(a) (i) Write down an expression for $\tan (2 x)$ in terms of $\tan x$

(ii) Hence show that $\tan (3 x)=\frac{3 \tan x-\tan ^{3} x}{1-3 \tan ^{2} x}$

Given that $\alpha$ is the acute angle such that $\cos \alpha=\frac{1}{3}$

(b) find the exact value of $\tan \alpha$ (2)

(c) Hence use the identity in part (a) to find the exact value of $\tan (3 \alpha)$

Give your answer in the form $\frac{a \sqrt{2}}{b}$ where $a$ and $b$ are integers. (2)


19. (2017/june/paper01/q9)

Using

$$\cos (A+B)=\cos A \cos B-\sin A \sin B$$

(a) show that $\cos ^{2} \theta=\frac{1}{2}(\cos 2 \theta+1)$

$$f(\theta)=8 \cos ^{4} \theta+4 \cos ^{2} \theta-5$$

Hence

(c) solve, for $0^{\circ} \leqslant x<180^{\circ}$, the equation

$$8 \cos ^{4} x+4 \cos ^{2} x-6 \cos 2 x=4.5$$

(d) find

(i) $\displaystyle\int \mathrm{f}(\theta) \mathrm{d} \theta$

(ii) the exact value of $\displaystyle\int_{0}^{\frac{\pi}{3}} f(\theta) \mathrm{d} \theta$


20. (2018/jan/paper01/q10)

$$\cos (A+B)=\cos A \cos B-\sin A \sin B$$

(a) Show that $\cos ^{2} \theta=\frac{1}{2}(\cos 2 \theta+1)$

Given that $f(\theta)=8 \cos ^{4} \theta+8 \sin ^{2} \theta-7$

(b) show that $\mathrm{f}(\theta)=\cos 4 \theta$

(c) Solve, for $0 \leqslant \theta \leqslant \frac{\pi}{2}$, the equation

$$16 \cos ^{4}\left(\theta-\frac{\pi}{6}\right)+16 \sin ^{2}\left(\theta-\frac{\pi}{6}\right)-15=0$$

(d) Using calculus, find the exact value of

$$\displaystyle\int_{0}^{\frac{\pi}{2}}\left(8 \cos ^{4} \theta+8 \sin ^{2} \theta+2 \sin 2 \theta\right) \mathrm{d} \theta$$


21. (2018/june/рарег02/q8)

$$\begin{aligned}&\cos (A+B)=\cos A \cos B-\sin A \sin B \\&\sin (A+B)=\sin A \cos B+\cos A \sin B\end{aligned}$$

Using the above identities

(a) show that

(i) $\cos 2 \theta=1-2 \sin ^{2} \theta$

(ii) $\sin 2 \theta=2 \sin \theta \cos \theta$

$$\mathrm{f}(\theta)=\cos 4 \theta+2 \cos 2 \theta$$

(b) Show that $\mathrm{f}(\theta)=8 \sin ^{4} \theta-12 \sin ^{2} \theta+3$ (4)

(c) Solve, giving your solutions to 3 significant figures, the equation

$$4 \sin ^{4} x^{\circ}-6 \sin ^{2} x^{\circ}-\cos 2 x^{\circ}+1.2=0 \quad 0 \leqslant x<90$$

(d) (i) Find $\displaystyle\int\left(2 \sin ^{4} \theta-3 \sin ^{2} \theta\right) \mathrm{d} \theta$

(ii) Hence find the exact value of $\displaystyle\int_{0}^{\frac{\pi}{3}}\left(2 \sin ^{4} \theta-3 \sin ^{2} \theta\right) \mathrm{d} \theta$

Give your answer in the form $a \sqrt{b}-c \pi$ where $a$ and $c$ are rational numbers and $b$ is a prime number.


22. (2019/june/рарег01/q7)

(a) Solve, in degrees to one decimal place,

$$(3 \cos \theta+5)(5 \sin \theta-3)=0 \quad \text { for } 0 \leqslant \theta<180^{\circ} $$

(b) Show that the equation

$$8 \sin (x-\alpha)=3 \sin (x+a)$$

can be written in the form

$$5 \tan x=11 \tan \alpha$$

(c) Hence solve, to one decimal place,

$$8 \sin \left(2 y-30^{\circ}\right)=3 \sin \left(2 y+30^{\circ}\right) \quad \text { for } 0 \leqslant y<180^{\circ}$$


23. (2019/juneR/paper02/q10)

(a) Use the formula for $\cos (A+B)$ to show that $\cos 2 A=2 \cos ^{2} A-1$ (2)

(b) Show that $\cos 4 A=8 \cos ^{4} A-8 \cos ^{2} A+1$

(c) Solve the equation $\cos ^{2}\left(\frac{\theta}{4}+\frac{\pi}{24}\right)\left[\cos ^{2}\left(\frac{\theta}{4}+\frac{\pi}{24}\right)-1\right]=-\frac{1}{16} \quad 0 \leqslant \theta<2 \pi$

Give your answers in terms of $\pi$.

$$\mathrm{f}(A)=4 \cos ^{4} A-4 \cos ^{2} A+1$$

(d) Using calculus, find the exact value of $\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \mathrm{f}(A) \mathrm{d} A$

Give your answer in the form $a \pi-b \sqrt{c}$ where $a$ and $b$ are fractions in their lowest terms and $c$ is a prime number.


24. (2011/june/paper01/q4)

$$\begin{aligned}&\sin (A+B)=\sin A \cos B+\cos A \sin B \\&\cos (A+B)=\cos A \cos B-\sin A \sin B\end{aligned}$$

(a) Write down an expression for $\sin 2 A$ in terms of $\sin A$ and $\cos A$

(b) Find an expression for $\cos 2 A$ in terms of $\sin A$(2)

(c) Show that $\sin 3 A+\sin A=4 \sin A-4 \sin ^{3} A$



Answer

1.(a) Show (b) Show

2. (a) $\theta=1.249,2.034$ (b) $\theta=0.3398,2.802$

3.(a) Show (b) $\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$ (c) show (d) show (e) $-2 \pm \sqrt{3}$

4.(a) $\frac{1}{\sqrt{2}} \cos 2 x-\frac{1}{\sqrt{2}} \sin 2 x$ (b) $x=0,135^{\circ}, 180^{\circ}, \ldots$ (c) $k=\sqrt{2}$ (d) $157.5^{\circ}$

5.(a) $\theta=0.201,2.94$ (b) $\theta=1.24,2.81$ (c) $\theta=1.32$

6. $x=-60^{\circ}, 60^{\circ}$

7.(a) Show (b) $y=24.9, 84.9^{\circ}, 144.9^{\circ}$

8.(a) $\theta=28.2^{\circ}, 118.2^{\circ}$ (b) $\theta=146.4^{\circ}$

9.(a) (i) Show (ii) $2 \sin A \cos A$ (b) Show (c) $x=0.0842,0.963,2.18,3.06$ (d) $-18 \cos \theta+2 \cos 3 \theta+6 \sin \theta+c$ (e) $5+3 \sqrt{3}$

10.(a) Show (b)show (c) $\theta=45^{\circ}, 123.7^{\circ}$

11.(a) $\frac{1}{\sqrt{2}}$ (b) Show (c) $\frac{\sqrt{3}-\sqrt{5}}{4}$ (d) Show

12.(a)(i) Show (ii) $2 \sin A \cos A$ (b) show (c) $A=70^{\circ},-10^{\circ},-5$ (d)(i) $\frac{1}{4}\left(-3 \cos \theta+\frac{1}{3} \cos 3 \theta\right)$ (ii) $\frac{8-5 \sqrt{2}}{12}$

13.(a) $x=1.159, x=-1.159$ (b) $\theta=0.099,1.669$

14. Show

15.(a) Show (b) $y=123^{\circ}$

16.(a) Show (b) $2\sin\theta\cos\theta$ (c) Show (d) $\theta=\frac{\pi}{9}, \frac{5 \pi}{9}, \frac{7 \pi}{9}$ $(e)(i) \frac{2}{3} \sin 3 \theta+6 \sin \theta-4 \cos \theta+C$ (ii) $3 \sqrt{3}+2$

17.(a) Show (b) $\theta=60.3,85$

18. $(a)(i) \tan (2 x)=\frac{2 \tan x}{1-\tan ^{2} x} \quad$ (ii) Show (b) $2 \sqrt{2}$ (c) $\frac{10 \sqrt{2}}{23}$

19. (a) show (b) Show (c) $x=30,60,120,150$ (d) (i) $\frac{1}{4} \sin 4 \theta+3 \sin 2 \theta+c$ (ii) $\frac{11 \sqrt{3}}{8}$

20.(a) Show (b) Show (c) $\frac{\pi}{4}, \frac{\pi}{12}$ (d) 2+$\frac{7\pi}{2}$

21.(a) Show (b) Show (c) $x=13.3^{\circ},76.7^{\circ}$ (d)(i) $\frac{1}{4}(\frac{1}{4}\sin 4\theta+\sin2\theta-3\theta)+C$ (ii) $\frac{3}{32}\sqrt{3}-\frac{\pi}{4}$

22. (a) $\theta=36.9^{\circ}$ or $143.1$ (b) Show (c) $y=25.9$ or $115.9$

23.(a) Show (b) Show (c) $\theta=\frac{\pi}{6},\frac{3\pi}{2}$ (d) $\frac{\pi}{6}-\frac{\sqrt 3}{16}$

24.(a) $2 \sin A \cos A$ (b) $1-2 \sin ^{2} A$ (c) Show


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